Source video: 【俗说矩阵】初等矩阵可一点也不初等，它居然有这么重要的意义！
3 elementary row operations
- 换行：交换两行的位置
- 数乘：给某一行乘以非零常数k
- 倍加：将某一行的 k 倍加到另一行上
By means of these 3 kind of elementary row operations, a matrix can be transformed to row echelon form, which is useful in analysing and solving the system of linear equations.
Identical matrix
$$ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$
The elements on the main diagonal are all 1 and the rest of elements in the martrix are all 0.
If multiplying a matrix with a identical matrix, the matrix doesn’t change. So the effect of I is similar to the 1 in numbers.
A identical matrix can perform elementary row operations.
Elementary matrix
对单位矩阵 I 实施一次初等行变换。实施2次就不是了。wikipedia
$$ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} ➔ \begin{cases} \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} & \text{Row switch: r1 ⟷ r2} \\ \\ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 1 \end{bmatrix} & \text{Row multiplication: 3r1 ➔ r1} \\ \\ \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} & \text{Row addition: 3r1+r2 ➔ r2} \end{cases} $$
初等矩阵只有3种：置换阵，数乘阵，倍加阵
Elementary matrices connect the matrix multiplication and elementary row operations.
Row-switching transformations
Row-multiplying transformations
Row-addition transformations
对矩阵实施一次初等行变换
左乘一个初等矩阵，就是把对初等矩阵的初等行变换，施加到矩阵上。